PROPAGATION OF LONGITUDINAL WAVES IN A MINDLIN-HERRMANN ROD IMMERSED IN A NONLINEAR ELASTIC MEDIUM

In this work, the propagation of longitudinal waves in a homogeneous rod immersed in a non-linear elastic medium is studied. The dynamic behavior of the rod is determined by the Mindlin – Herrmann theory, which neglects the hypothesis of the proportionality of transverse deformations to longitudinal deformations under axial tension or compression. To describe the movement of particles in the transverse direction, an additional function is introduced that provides greater model accuracy. The initial system of equations reduces to one nonlinear equation of the fourth order relative to the longitudinal displacement of the rod particles. This equation, on the one hand, allows you to obtain an evolutionary equation and find its exact solution, on the other hand, it allows qualitative research in two special cases in traveling wave variables. It is shown that the evolutionary equation is Ostrovsky's equation with an additional quadratic-nonlinear term. By the method of the simplest equations for the evolutionary equation, exact solutions were found from the class of stationary waves that retain their shape and speed during propagation. Waves have the appearance of a soliton of a classic profile. The type of non-linearity (soft, hard) of the medium affects the polarity of the localized wave. A medium with soft nonlinearity corresponds to a soliton of positive polarity. Relationships of amplitude, width and velocity of nonlinear wave from system parameters characterizing nonlinear elastic medium are obtained. Cases available for qualitative research are possible if the coefficient at the highest derivative is equal to zero. In one case, the equation of an anharmonic oscillator follows, with two types of quadratic nonlinearity. The first integral of the equation was obtained and phase portraits were constructed at various ratios of system parameters affecting the existence and appearance of phase trajectories. In another case, the classical equation of an anharmonic oscillator with quadratic nonlinearity follows, which is quite well studied. Qualitative analysis of particular cases shows the possibility of the existence, in the system under consideration, of localized and nonlinear periodic waves.